Set Theory

Choice

Equation form expr-007d3ba7b1c1b0d3

⋃A\bigcup A

Read as: the union of capital A

Means: the union of capital A

4 occurrences in this chapter

Equation form expr-0198615a697deaf3

S\onesphere

Read as: the unit circle

Means: the unit circle

13 occurrences in this chapter

Equation form expr-03b8daa53ab33876

[0,2π)[0,2\pi)

Read as: the half open interval from zero inclusive to two pi exclusive

Means: the half open interval from zero inclusive to two pi exclusive

1 occurrence in this chapter

Equation form expr-046d6e2692196af0

{x∈Vα+1:ϕ(x)}\Setabs{x \in V_{\alpha+1}}{\phi(x)}

Read as: the set of x belongs to capital V subscript alpha plus one such that phi of x

Means: the set of x belongs to capital V subscript alpha plus one such that phi of x

1 occurrence in this chapter

Equation form expr-05ab6a53ee04cbbc

{⟨a,b⟩∈A×A:f(a)∈f(b)}\Setabs{\tuple{a, b} \in A \times A}{f(a) \in f(b)}

Read as: the set of the ordered pair a, then b belongs to capital A times capital A such that f of a belongs to f of b

Means: the set of the ordered pair a, then b belongs to capital A times capital A such that f of a belongs to f of b

1 occurrence in this chapter

Equation form expr-063d17f3eb0b3826

ρ−1∈R\rho^{-1} \in \rotationsgroup

Read as: rho superscript minus one belongs to the group of rotations of the unit circle

Means: rho superscript minus one belongs to the group of rotations of the unit circle

1 occurrence in this chapter

Equation form expr-067b4661e2a64a54

c1,…,cnc_1, \ldots, c_n

Read as: c subscript one, then c subscript n, and so on

Means: c subscript one, then c subscript n, and so on

1 occurrence in this chapter

Equation form expr-069793e854be09a7

[x:ϕ(x)][ x : \phi(x)]

Read as: the Tarski Scott set of all x of least possible rank such that phi holds of x, or the empty set if there are none

Means: the Tarski Scott set of all x of least possible rank such that phi holds of x, or the empty set if there are none

1 occurrence in this chapter

Equation form expr-08f271887ce94707

MM

Read as: capital M

Means: capital M

1 occurrence in this chapter

Equation form expr-099959186b7f6cac

n=mn = m

Read as: n equals m

Means: n equals m

1 occurrence in this chapter

Equation form expr-09d54469d5b38cbe

μ(E)∈ℝ\mu(E) \in \Real

Read as: mu of capital E belongs to the real numbers

Means: mu of capital E belongs to the real numbers

1 occurrence in this chapter

Equation form expr-0b94fa72b4f7fcf5

f(x)∈xf(x) \in x

Read as: f of x belongs to x

Means: f of x belongs to x

1 occurrence in this chapter

Equation form expr-0e4763978622e187

f1f_1

Read as: f subscript one

Means: f subscript one

1 occurrence in this chapter

Equation form expr-0e663e0f57f57a74

n∈ωn \in \omega

Read as: n belongs to omega

Means: n belongs to omega

5 occurrences in this chapter

Equation form expr-10832ea38ded7f1f

D1D_{1}

Read as: capital D subscript one

Means: capital D subscript one

2 occurrences in this chapter

Equation form expr-10ce0200b450ea95

ϕ\phi

Read as: phi

Means: phi

2 occurrences in this chapter

Equation form expr-11baa595827a4e0f

ω\omega

Read as: omega

Means: omega

2 occurrences in this chapter

Equation form expr-152461375bf3fefd

A0,A1,A2,…A_0, A_1, A_2, \ldots

Read as: capital A subscript zero, then capital A subscript one, then capital A subscript two, and so on

Means: capital A subscript zero, then capital A subscript one, then capital A subscript two, and so on

1 occurrence in this chapter

Equation form expr-174d7a05108c32f9

α⋠A\cardnless{\alpha}{A}

Read as: alpha does not have smaller cardinality than capital A

Means: alpha does not have smaller cardinality than capital A

1 occurrence in this chapter

Equation form expr-17eacf555942d161

a={b1,…,bn}a = \{b_1, \ldots, b_n\}

Read as: a equals the set containing b subscript one, then , and so on, then b subscript n

Means: a equals the set containing b subscript one, then , and so on, then b subscript n

1 occurrence in this chapter

Equation form expr-1800ec380b7fab3f

g(α)=f(A∖g[α])∈A∖g[α]g(\alpha) = f(A \setminus \funimage{g}{\alpha}) \in A \setminus \funimage{g}{\alpha}

Read as: g of alpha equals f of capital A set minus the image of alpha under g belongs to capital A set minus the image of alpha under g

Means: g of alpha equals f of capital A set minus the image of alpha under g belongs to capital A set minus the image of alpha under g

1 occurrence in this chapter

Equation form expr-18f5384d58bcb1bb

YY

Read as: capital Y

Means: capital Y

1 occurrence in this chapter

Equation form expr-1964032c5c55c22c

f(n)=cnf(n) = c_n

Read as: f of n equals c subscript n

Means: f of n equals c subscript n

1 occurrence in this chapter

Equation form expr-1a7a389794d655aa

f:α→Af \colon \alpha \to A

Read as: f colon alpha maps to capital A

Means: f colon alpha maps to capital A

1 occurrence in this chapter

Equation form expr-1a8b8dbc0d0a486a

g(δ)=Ag(\delta) = A

Read as: g of delta equals capital A

Means: g of delta equals capital A

1 occurrence in this chapter

Equation form expr-1b16b1df538ba12d

nn

Read as: n

Means: n

2 occurrences in this chapter

Equation form expr-1e7dcb8f53ff128f

f(x)=Source fragment ends inside a text argument.thef(x) = \text{the

Read as: f of x equals the

Means: f of x equals the

1 occurrence in this chapter

Equation form expr-20bb4634e6c63f72

β∈a\beta \in \cardfont{a}

Read as: beta belongs to the cardinal a

Means: beta belongs to the cardinal a

1 occurrence in this chapter

Equation form expr-20ff5a7abb4a64cf

σ[C]\funimage{\sigma}{C}

Read as: the image of capital C under sigma

Means: the image of capital C under sigma

1 occurrence in this chapter

Equation form expr-2225b5a8bdecda32

x∈Ax \in A

Read as: x belongs to capital A

Means: x belongs to capital A

2 occurrences in this chapter

Equation form expr-230df3ec6f3ebabf

r=0r = 0

Read as: r equals zero

Means: r equals zero

1 occurrence in this chapter

Equation form expr-24bed2b8115326c2

A≼B\cardle{A}{B}

Read as: capital A has cardinality at most that of capital B

Means: capital A has cardinality at most that of capital B

3 occurrences in this chapter

Equation form expr-252f10c83610ebca

ff

Read as: f

Means: f

7 occurrences in this chapter

Equation form expr-259072e6c653cc56

f(x)f(x)

Read as: f of x

Means: f of x

1 occurrence in this chapter

Equation form expr-26a48c8256d0b3ee

f(m,n)=fn(m)f(m, n) = f_n(m)

Read as: f of m and n equals f subscript n of m

Means: f of m and n equals f subscript n of m

1 occurrence in this chapter

Equation form expr-28b6564cf8eaf691

r>0r > 0

Read as: r is greater than zero

Means: r is greater than zero

1 occurrence in this chapter

Equation form expr-2952887bc45642d0

(0,1)(0,1)

Read as: open scope, zero, then one, close scope

Means: open scope, zero, then one, close scope

1 occurrence in this chapter

Equation form expr-2a4f64b55f230d6b

g(α)=g(β)g(\alpha) = g(\beta)

Read as: g of alpha equals g of beta

Means: g of alpha equals g of beta

1 occurrence in this chapter

Equation form expr-2a7afef6dabb7f09

σ∈R\sigma \in \rotationsgroup

Read as: sigma belongs to the group of rotations of the unit circle

Means: sigma belongs to the group of rotations of the unit circle

1 occurrence in this chapter

Equation form expr-2ae25ae35ecdcc89

C=ran(f)C = \ran{f}

Read as: capital C equals the range of f

Means: capital C equals the range of f

1 occurrence in this chapter

Equation form expr-2b7093d328352d52

tan(π(r−12)))\tan(\pi(r-\nicefrac{1}{2})))

Read as: the tangent of pi times the quantity r minus one half; the source formula contains one extra closing parenthesis

Means: the tangent of pi times the quantity r minus one half; the source formula contains one extra closing parenthesis

1 occurrence in this chapter

Equation form expr-2d05410d75295f5c

ω→A\omega \to A

Read as: omega maps to capital A

Means: omega maps to capital A

1 occurrence in this chapter

Equation form expr-2d711642b726b044

xx

Read as: x

Means: x

3 occurrences in this chapter

Equation form expr-2de1d0c7fd40518d

ρ∈C\rho \in C

Read as: rho belongs to capital C

Means: rho belongs to capital C

2 occurrences in this chapter

Equation form expr-2e103ad192990365

ρ2−1∘ρ1∈R\comp{\rho_1}{\rho^{-1}_2} \in \rotationsgroup

Read as: rho superscript minus one subscript two composed with rho subscript one belongs to the group of rotations of the unit circle

Means: rho superscript minus one subscript two composed with rho subscript one belongs to the group of rotations of the unit circle

1 occurrence in this chapter

Equation form expr-2e7ca95a7c902b9c

g(0)=f(A)g(α)={stop!if A=g[α]f(A∖g[α])otherwiseg(0) &= f(A)\\ g(\alpha) &= \begin{cases} \text{stop!{}} &\text{if }A = \funimage{g}{\alpha}\\ f(A \setminus \funimage{g}{\alpha}) & \text{otherwise}\\ \end{cases}

Read as: Source-ordered display. g of zero equals f of capital A. Then, g of alpha. End display is defined by cases. case one, stop!; if, capital A equals the image of alpha under g. case two, f of capital A set minus the image of alpha under g; otherwise. End cases

Means: Source-ordered display. g of zero equals f of capital A. Then, g of alpha. End display is defined by cases. case one, stop!; if, capital A equals the image of alpha under g. case two, f of capital A set minus the image of alpha under g; otherwise. End cases

1 occurrence in this chapter

Equation form expr-2ec1aad3554cc045

f:ω×ω→⋃n<ωAnf \colon \omega \times \omega \to \bigcup_{n < \omega} A_n

Read as: f colon omega times omega maps to the union over n is less than omega of capital A subscript n

Means: f colon omega times omega maps to the union over n is less than omega of capital A subscript n

1 occurrence in this chapter

Equation form expr-301bd5aa9c52d957

ρ∈R\rho \in \rotationsgroup

Read as: rho belongs to the group of rotations of the unit circle

Means: rho belongs to the group of rotations of the unit circle

5 occurrences in this chapter

Equation form expr-33a5d7431cda0f1c

∑ρ∈Rr=∞\sum_{\rho \in \rotationsgroup}r = \infty

Read as: the sum over rho belongs to the group of rotations of the unit circle of r equals infinity

Means: the sum over rho belongs to the group of rotations of the unit circle of r equals infinity

1 occurrence in this chapter

Equation form expr-3588a61df3c7b068

α∈α\alpha \in \alpha

Read as: alpha belongs to alpha

Means: alpha belongs to alpha

1 occurrence in this chapter

Equation form expr-363435b5ce5f64a2

β⊆α\beta \subseteq \alpha

Read as: beta is a subset of alpha

Means: beta is a subset of alpha

2 occurrences in this chapter

Equation form expr-37e9a9961f225268

A≈n\cardeq{A}{n}

Read as: capital A is equinumerous with n

Means: capital A is equinumerous with n

2 occurrences in this chapter

Equation form expr-3d6380742c628b38

D2D_2

Read as: capital D subscript two

Means: capital D subscript two

3 occurrences in this chapter

Equation form expr-3d7d0253a8ca9a09

fβf_\beta

Read as: f subscript beta

Means: f subscript beta

1 occurrence in this chapter

Equation form expr-3f25ed8a41612223

c1∈b1,…,cn∈bnc_1 \in b_1, \ldots, c_n \in b_n

Read as: c subscript one belongs to b subscript one, then c subscript n, and so on belongs to b subscript n

Means: c subscript one belongs to b subscript one, then c subscript n, and so on belongs to b subscript n

1 occurrence in this chapter

Equation form expr-4288d603a0dca00d

bi≠∅b_i \neq \emptyset

Read as: b subscript i is not equal to the empty set

Means: b subscript i is not equal to the empty set

1 occurrence in this chapter

Equation form expr-440c9fc4b6198138

δ≤α\delta \leq \alpha

Read as: delta is less than or equal to alpha

Means: delta is less than or equal to alpha

1 occurrence in this chapter

Equation form expr-454349e422f05297

rr

Read as: r

Means: r

1 occurrence in this chapter

Equation form expr-4553b7f3532fddef

⟨A,R⟩\tuple{A, R}

Read as: the ordered pair capital A, then capital R

Means: the ordered pair capital A, then capital R

1 occurrence in this chapter

Equation form expr-45d4e2b070f4de22

b∈Bb \in B

Read as: b belongs to capital B

Means: b belongs to capital B

1 occurrence in this chapter

Equation form expr-47b34af59b56e555

A1A_1

Read as: capital A subscript one

Means: capital A subscript one

1 occurrence in this chapter

Equation form expr-47bfc304486e05f3

g(α)∉g[α]g(\alpha) \notin \funimage{g}{\alpha}

Read as: g of alpha does not belong to the image of alpha under g

Means: g of alpha does not belong to the image of alpha under g

1 occurrence in this chapter

Equation form expr-481ed37aa799891c

g(β)∉g[α]g(\beta) \notin \funimage{g}{\alpha}

Read as: g of beta does not belong to the image of alpha under g

Means: g of beta does not belong to the image of alpha under g

1 occurrence in this chapter

Equation form expr-4823e6a7820d9b9f

A⊔B≈A×B≈M\cardeq{\cardeq{A \disjointsum B}{A \times B}}{M}

Read as: capital A disjoint sum capital B is equinumerous with capital A times capital B is equinumerous with capital M

Means: capital A disjoint sum capital B is equinumerous with capital A times capital B is equinumerous with capital M

1 occurrence in this chapter

Equation form expr-4893e9df8b5496eb

α\alpha

Read as: alpha

Means: alpha

6 occurrences in this chapter

Equation form expr-492247e1bab15272

μ\mu

Read as: mu

Means: mu

2 occurrences in this chapter

Equation form expr-4b5712fc1047872a

{⟨b1,c1⟩,…,⟨bn,cn⟩}\{\langle b_1, c_1\rangle , \ldots, \langle b_n, c_n\rangle\}

Read as: the set containing the ordered pair whose first component is b subscript one and whose second component is c subscript one, and so on through the ordered pair whose first component is b subscript n and whose second component is c subscript n

Means: the set containing the ordered pair whose first component is b subscript one and whose second component is c subscript one, and so on through the ordered pair whose first component is b subscript n and whose second component is c subscript n

1 occurrence in this chapter

Equation form expr-4b68ab3847feda7d

XX

Read as: capital X

Means: capital X

3 occurrences in this chapter

Equation form expr-4b7ce6b75f0ec36f

ϕ(x)\phi(x)

Read as: phi of x

Means: phi of x

2 occurrences in this chapter

Equation form expr-4c9fcda7cdb10255

R\rotationsgroup

Read as: the group of rotations of the unit circle

Means: the group of rotations of the unit circle

9 occurrences in this chapter

Equation form expr-4e9026ffa6659535

A≺B\cardless{A}{B}

Read as: capital A has smaller cardinality than capital B

Means: capital A has smaller cardinality than capital B

1 occurrence in this chapter

Equation form expr-522df6ad7d1bb5ca

Z−\Zminus

Read as: set theory Z minus

Means: set theory Z minus

1 occurrence in this chapter

Equation form expr-5280f599b29a1e30

ran(g)=A\ran{g} = A

Read as: the range of g equals capital A

Means: the range of g equals capital A

1 occurrence in this chapter

Equation form expr-53734baf3eedc67e

tsc(A)=[x:A≈x]\text{tsc}(A) = [x : \cardeq{A}{x}]

Read as: the T S cardinality of capital A equals the Tarski Scott set of all x of least possible rank that are equinumerous with capital A

Means: the T S cardinality of capital A equals the Tarski Scott set of all x of least possible rank that are equinumerous with capital A

1 occurrence in this chapter

Equation form expr-5457674e632558b5

α⊆β\alpha \subseteq \beta

Read as: alpha is a subset of beta

Means: alpha is a subset of beta

2 occurrences in this chapter

Equation form expr-554fc80c26fb20e3

S=⋃ρ∈Rρ[C]\onesphere = \bigcup_{\rho \in \rotationsgroup} \funimage{\rho}{C}

Read as: the unit circle equals the union over rho belongs to the group of rotations of the unit circle of the image of capital C under rho

Means: the unit circle equals the union over rho belongs to the group of rotations of the unit circle of the image of capital C under rho

1 occurrence in this chapter

Equation form expr-559aead08264d579

AA

Read as: capital A

Means: capital A

27 occurrences in this chapter

Equation form expr-583e90eb7865276c

B⊆AB \subseteq A

Read as: capital B is a subset of capital A

Means: capital B is a subset of capital A

1 occurrence in this chapter

Equation form expr-58d0485ca992db83

μ(S)=1\mu(\onesphere) = 1

Read as: mu of the unit circle equals one

Means: mu of the unit circle equals one

2 occurrences in this chapter

Equation form expr-59440755b9d63ad2

R2=R∖R1\rotationsgroup_{2} = \rotationsgroup \setminus \rotationsgroup_1

Read as: the group of rotations of the unit circle subscript two equals the group of rotations of the unit circle set minus the group of rotations of the unit circle subscript one

Means: the group of rotations of the unit circle subscript two equals the group of rotations of the unit circle set minus the group of rotations of the unit circle subscript one

1 occurrence in this chapter

Equation form expr-5b6c985d29cce910

ω×ω\omega \times \omega

Read as: omega times omega

Means: omega times omega

1 occurrence in this chapter

Equation form expr-5c0922e4ca62ba72

ρ,σ,τ∈R\rho, \sigma, \tau \in \rotationsgroup

Read as: rho, then sigma, then tau belongs to the group of rotations of the unit circle

Means: rho, then sigma, then tau belongs to the group of rotations of the unit circle

1 occurrence in this chapter

Equation form expr-5c1715c895704541

{x:A≈x}\Setabs{x}{\cardeq{A}{x}}

Read as: the set of x such that capital A is equinumerous with x

Means: the set of x such that capital A is equinumerous with x

1 occurrence in this chapter

Equation form expr-5d6e77705844ff82

⋃n<ωAn\bigcup_{n < \omega} A_n

Read as: the union over n is less than omega of capital A subscript n

Means: the union over n is less than omega of capital A subscript n

1 occurrence in this chapter

Equation form expr-5feceb66ffc86f38

00

Read as: zero

Means: zero

1 occurrence in this chapter

Equation form expr-6049811188f66e4a

(τ∘σ)∘ρ=τ∘(σ∘ρ)\comp{\rho}{(\comp{\sigma}{\tau})} = \comp{(\comp{\rho}{\sigma})}{\tau}

Read as: open scope, tau composed with sigma, close scope composed with rho equals tau composed with open scope, sigma composed with rho, close scope

Means: open scope, tau composed with sigma, close scope composed with rho equals tau composed with open scope, sigma composed with rho, close scope

1 occurrence in this chapter

Equation form expr-616522c60d7b122c

∼\sim

Read as: is equivalent to

Means: is equivalent to

3 occurrences in this chapter

Equation form expr-61b55354f7616dbb

β∉α\beta \notin \alpha

Read as: beta does not belong to alpha

Means: beta does not belong to alpha

1 occurrence in this chapter

Equation form expr-6283787855819935

R2\rotationsgroup_{2}

Read as: the group of rotations of the unit circle subscript two

Means: the group of rotations of the unit circle subscript two

2 occurrences in this chapter

Equation form expr-63984962c330f4f1

n<ωn < \omega

Read as: n is less than omega

Means: n is less than omega

1 occurrence in this chapter

Equation form expr-64c91d1f48514310

{ρ∘ρ:ρ∈R1}=R\Setabs{\comp{\rho}{\rho}}{\rho \in \rotationsgroup_1} = \rotationsgroup

Read as: the set of rho composed with rho such that rho belongs to the group of rotations of the unit circle subscript one, equals the group of rotations of the unit circle

Means: the set of rho composed with rho such that rho belongs to the group of rotations of the unit circle subscript one, equals the group of rotations of the unit circle

1 occurrence in this chapter

Equation form expr-64f11df8fdf81250

2n\cardexpo{2}{n}

Read as: cardinal exponentiation of two to the power n

Means: cardinal exponentiation of two to the power n

1 occurrence in this chapter

Equation form expr-669ac118248289e5

|A|\card{A}

Read as: the cardinality of capital A

Means: the cardinality of capital A

1 occurrence in this chapter

Equation form expr-671ea787d9a0d6c3

γ∈β∈α\gamma \in \beta \in \alpha

Read as: gamma belongs to beta belongs to alpha

Means: gamma belongs to beta belongs to alpha

1 occurrence in this chapter

Equation form expr-68a9d5a8339e4038

cnc_n

Read as: c subscript n

Means: c subscript n

2 occurrences in this chapter

Equation form expr-6aea76f4b6917569

f:A→βf \colon A \to \beta

Read as: f colon capital A maps to beta

Means: f colon capital A maps to beta

1 occurrence in this chapter

Equation form expr-6b23c0d5f35d1b11

CC

Read as: capital C

Means: capital C

5 occurrences in this chapter

Equation form expr-6b86b273ff34fce1

11

Read as: one

Means: one

2 occurrences in this chapter

Equation form expr-6deda92c5c095677

ρ,σ∈R\rho, \sigma \in \rotationsgroup

Read as: rho, then sigma belongs to the group of rotations of the unit circle

Means: rho, then sigma belongs to the group of rotations of the unit circle

1 occurrence in this chapter

Equation form expr-6e0b6a45eec54055

12n\nicefrac{1}{2^n}

Read as: the fraction one over two superscript n

Means: the fraction one over two superscript n

1 occurrence in this chapter

Equation form expr-6e402c0e6cf585b8

2π−r2\pi - r

Read as: two pi minus r

Means: two pi minus r

1 occurrence in this chapter

Equation form expr-6e82ac27792a9ac6

ℝ\Real

Read as: the real numbers

Means: the real numbers

1 occurrence in this chapter

Equation form expr-6e9c4d9fd68b99dd

μ(ρ[C])=r\mu(\funimage{\rho}{C}) = r

Read as: mu of the image of capital C under rho equals r

Means: mu of the image of capital C under rho equals r

1 occurrence in this chapter

Equation form expr-6f9db049b056a5c3

A=g[α]A = g[\alpha]

Read as: capital A equals the image of alpha under g

Means: capital A equals the image of alpha under g

1 occurrence in this chapter

Equation form expr-7046857daf500937

ZF\ZF

Read as: set theory Z F

Means: set theory Z F

9 occurrences in this chapter

Equation form expr-70f34815ff612b32

ρ∘0R=0R∘ρ=ρ\comp{0_{\rotationsgroup}}{\rho} = \comp{\rho}{0_{\rotationsgroup}} = \rho

Read as: rho composed with zero subscript the group of rotations of the unit circle equals zero subscript the group of rotations of the unit circle composed with rho

Means: rho composed with zero subscript the group of rotations of the unit circle equals zero subscript the group of rotations of the unit circle composed with rho

1 occurrence in this chapter

Equation form expr-71a45cef8ce2d204

[0,π)[0, \pi)

Read as: the half open interval from zero inclusive to pi exclusive

Means: the half open interval from zero inclusive to pi exclusive

1 occurrence in this chapter

Equation form expr-751dcdf272a503d8

R2\rotationsgroup_2

Read as: the group of rotations of the unit circle subscript two

Means: the group of rotations of the unit circle subscript two

1 occurrence in this chapter

Equation form expr-7577f453de17bcbc

{⟨r,s⟩∈ℝ2:r2+s2=1}\Setabs{\tuple{r,s} \in \Real^2}{\sqrt{r^2+s^2}=1}

Read as: the set of ordered pairs r and s in the Cartesian square of the real numbers such that the square root of r squared plus s squared equals one

Means: the set of ordered pairs r and s in the Cartesian square of the real numbers such that the square root of r squared plus s squared equals one

1 occurrence in this chapter

Equation form expr-76ee45c761ac24bc

r∈[s]∼r \in \equivrep{s}{\sim}

Read as: r belongs to the equivalence class of s modulo is equivalent to

Means: r belongs to the equivalence class of s modulo is equivalent to

1 occurrence in this chapter

Equation form expr-773422fda490df28

⟨B,R⟩⊆Vrank(A)+4\tuple{B, R} \subseteq V_{\setrank{A}+4}

Read as: the ordered pair capital B, then capital R is a subset of capital V subscript the rank of capital A plus four

Means: the ordered pair capital B, then capital R is a subset of capital V subscript the rank of capital A plus four

1 occurrence in this chapter

Equation form expr-797cf7ed6930fb72

g:β→⟨B,S⟩g \colon \beta \to \tuple{B, S}

Read as: g colon beta maps to the ordered pair capital B, then capital S

Means: g colon beta maps to the ordered pair capital B, then capital S

1 occurrence in this chapter

Equation form expr-7dfd61dc348d13af

(∃x⊆Vα)ϕ(x)(\exists x\subseteq V_\alpha)\phi(x)

Read as: there exists x is a subset of capital V subscript alpha, phi of x

Means: there exists x is a subset of capital V subscript alpha, phi of x

1 occurrence in this chapter

Equation form expr-7e3fb797657350ce

γ∈α\gamma \in \alpha

Read as: gamma belongs to alpha

Means: gamma belongs to alpha

1 occurrence in this chapter

Equation form expr-7f6aa69dadf31d5d

0R0_{\rotationsgroup}

Read as: zero subscript the group of rotations of the unit circle

Means: zero subscript the group of rotations of the unit circle

1 occurrence in this chapter

Equation form expr-7fd96886f4590d66

dom(f)=A∖{∅}\dom{f} = A \setminus \{\emptyset\}

Read as: the domain of f equals capital A set minus the set containing the empty set

Means: the domain of f equals capital A set minus the set containing the empty set

1 occurrence in this chapter

Equation form expr-81e04c922d77656e

s∈ρ1[C]∩ρ2[C]s \in \funimage{\rho_{1}}{C} \cap \funimage{\rho_{2}}{C}

Read as: s belongs to the image of capital C under rho subscript one intersected with the image of capital C under rho subscript two

Means: s belongs to the image of capital C under rho subscript one intersected with the image of capital C under rho subscript two

1 occurrence in this chapter

Equation form expr-89a67f5ba6161b7b

D1=⋃ρ∈R1ρ[C]D2=⋃ρ∈R2ρ[C]D_{1} &= \bigcup_{\rho \in \rotationsgroup_1} \funimage{\rho}{C} & D_{2} &= \bigcup_{\rho \in \rotationsgroup_2} \funimage{\rho}{C}

Read as: capital D subscript one equals the union over rho belongs to the group of rotations of the unit circle subscript one of the image of capital C under rho capital D subscript two equals the union over rho belongs to the group of rotations of the unit circle subscript two of the image of capital C under rho

Means: capital D subscript one equals the union over rho belongs to the group of rotations of the unit circle subscript one of the image of capital C under rho capital D subscript two equals the union over rho belongs to the group of rotations of the unit circle subscript two of the image of capital C under rho

1 occurrence in this chapter

Equation form expr-8aacb74ff69eb254

ρ∘ρ−1=0R\rho \circ \rho^{-1} = 0_\rotationsgroup

Read as: rho composed with rho superscript minus one equals zero subscript the group of rotations of the unit circle

Means: rho composed with rho superscript minus one equals zero subscript the group of rotations of the unit circle

1 occurrence in this chapter

Equation form expr-8c20324c840bb59b

R⊆B2R \subseteq B^2

Read as: capital R is a subset of capital B superscript two

Means: capital R is a subset of capital B superscript two

1 occurrence in this chapter

Equation form expr-8c49fbee10c92748

⟨A,<⟩\tuple{A, <}

Read as: the ordered pair capital A, then is less than

Means: the ordered pair capital A, then is less than

1 occurrence in this chapter

Equation form expr-8cafa9e3d0a185fe

ℵ0\aleph_0

Read as: aleph subscript zero

Means: aleph subscript zero

2 occurrences in this chapter

Equation form expr-8d2cacefc75ba038

∅\emptyset

Read as: the empty set

Means: the empty set

1 occurrence in this chapter

Equation form expr-8de0b3c47f112c59

SS

Read as: capital S

Means: capital S

8 occurrences in this chapter

Equation form expr-90bbf12a76c6e3e1

α={ord(B,R):⟨B,R⟩∈C}.\alpha = \Setabs{\ordtype{B, R}}{\tuple{B, R} \in C}.

Read as: alpha equals the set of the order type of capital B, then capital R such that the ordered pair capital B, then capital R belongs to capital C

Means: alpha equals the set of the order type of capital B, then capital R such that the ordered pair capital B, then capital R belongs to capital C

1 occurrence in this chapter

Equation form expr-9124961853879c66

An≠∅A_n \neq \emptyset

Read as: capital A subscript n is not equal to the empty set

Means: capital A subscript n is not equal to the empty set

1 occurrence in this chapter

Equation form expr-9204fc9b17bec102

cn=cmc_n = c_m

Read as: c subscript n equals c subscript m

Means: c subscript n equals c subscript m

1 occurrence in this chapter

Equation form expr-95025b55ecf3ff52

|⋃i<nAn|≤|A0|+|A1|+…+|An−1|=1+2+…+2n−1=2n−1<2n=|An|\card{\bigcup_{i < n}A_n} &\leq \card{A_0} + \card{A_1} + \ldots + \card{A_{n-1}}\\ &=1 + 2 + \ldots + 2^{n-1}\\ & = 2^n - 1\\ & < 2^n = \card{A_n}

Read as: Source-ordered display. the cardinality of the union over i is less than n of capital A subscript n is less than or equal to the cardinality of capital A subscript zero plus the cardinality of capital A subscript one. Then, equals one plus two plus , and so on plus two superscript n minus one. Then, equals two superscript n minus one. Then, is less than two superscript n equals the cardinality of capital A subscript n. End display

Means: Source-ordered display. the cardinality of the union over i is less than n of capital A subscript n is less than or equal to the cardinality of capital A subscript zero plus the cardinality of capital A subscript one. Then, equals one plus two plus , and so on plus two superscript n minus one. Then, equals two superscript n minus one. Then, is less than two superscript n equals the cardinality of capital A subscript n. End display

1 occurrence in this chapter

Equation form expr-98550483d35bb33d

f2f_2

Read as: f subscript two

Means: f subscript two

1 occurrence in this chapter

Equation form expr-993220b6cfa76019

A0A_0

Read as: capital A subscript zero

Means: capital A subscript zero

1 occurrence in this chapter

Equation form expr-99c4fa5eb137c004

C={⟨B,R⟩∈Vrank(A)+5:B⊆A and ⟨B,R⟩ is a well-ordering}C = \Setabs{\tuple{B, R} \in V_{\setrank{A}+5}}{B\subseteq A \text{ and $\tuple{B, R}$ is a well-ordering}}

Read as: capital C equals the set of the ordered pair capital B, then capital R belongs to capital V subscript the rank of capital A plus five such that capital B is a subset of capital A, and the ordered pair capital B, then capital R is a well-ordering

Means: capital C equals the set of the ordered pair capital B, then capital R belongs to capital V subscript the rank of capital A plus five such that capital B is a subset of capital A, and the ordered pair capital B, then capital R is a well-ordering

1 occurrence in this chapter

Equation form expr-99e5f8e078f9fe31

D1D_1

Read as: capital D subscript one

Means: capital D subscript one

3 occurrences in this chapter

Equation form expr-9c123d16cfd8af14

ρ1≠ρ2\rho_1 \neq \rho_2

Read as: rho subscript one is not equal to rho subscript two

Means: rho subscript one is not equal to rho subscript two

1 occurrence in this chapter

Equation form expr-9c3245dfb4ac54c1

σ\sigma

Read as: sigma

Means: sigma

1 occurrence in this chapter

Equation form expr-9c977bea2716b588

ρ[C]\funimage{\rho}{C}

Read as: the image of capital C under rho

Means: the image of capital C under rho

6 occurrences in this chapter

Equation form expr-a0f6e7d4d2798058

rank(x)\setrank{x}

Read as: the rank of x

Means: the rank of x

1 occurrence in this chapter

Equation form expr-a10c8fc3a5760311

α=β\alpha = \beta

Read as: alpha equals beta

Means: alpha equals beta

1 occurrence in this chapter

Equation form expr-a3fe53372150c2ba

r1=r2r_1 = r_2

Read as: r subscript one equals r subscript two

Means: r subscript one equals r subscript two

1 occurrence in this chapter

Equation form expr-a42f60b5a5872237

S=⋃ρ∈Rρ[C]\onesphere = \bigcup_{\rho \in \rotationsgroup}\funimage{\rho}{C}

Read as: the unit circle equals the union over rho belongs to the group of rotations of the unit circle of the image of capital C under rho

Means: the unit circle equals the union over rho belongs to the group of rotations of the unit circle of the image of capital C under rho

1 occurrence in this chapter

Equation form expr-a4796253fe566663

ρ2−1(ρ1(r1))=r2\rho^{-1}_2(\rho_1(r_1)) = r_2

Read as: rho superscript minus one subscript two of rho subscript one of r subscript one equals r subscript two

Means: rho superscript minus one subscript two of rho subscript one of r subscript one equals r subscript two

1 occurrence in this chapter

Equation form expr-a587b09c0270b677

μ(⋃n<ωXn)=∑n<ωμ(Xn)\mu(\bigcup_{n < \omega} X_n) = \sum_{n < \omega}\mu(X_n)

Read as: mu of the union over n is less than omega of capital X subscript n equals the sum over n is less than omega of mu of capital X subscript n

Means: mu of the union over n is less than omega of capital X subscript n equals the sum over n is less than omega of mu of capital X subscript n

1 occurrence in this chapter

Equation form expr-a70abc50c926afc2

∑ρ∈Rr=0\sum_{\rho \in \rotationsgroup}r = 0

Read as: the sum over rho belongs to the group of rotations of the unit circle of r equals zero

Means: the sum over rho belongs to the group of rotations of the unit circle of r equals zero

1 occurrence in this chapter

Equation form expr-a7211c782fe0d476

ℝ3\Real^3

Read as: the real numbers superscript three

Means: the real numbers superscript three

3 occurrences in this chapter

Equation form expr-a75dbfeaae9c1c65

f0f_0

Read as: f subscript zero

Means: f subscript zero

1 occurrence in this chapter

Equation form expr-a783986960bb8aa8

Source fragment starts inside a text argument.-least member of x-least member of }x

Read as: least member of x with respect to the preceding less than ordering

Means: least member of x with respect to the preceding less than ordering

1 occurrence in this chapter

Equation form expr-a7ae32501af31c2a

Z−+Countable Choice\Zminus + \text{Countable Choice}

Read as: set theory Z minus plus Countable Choice

Means: set theory Z minus plus Countable Choice

2 occurrences in this chapter

Equation form expr-a9f51566bd6705f7

EE

Read as: capital E

Means: capital E

3 occurrences in this chapter

Equation form expr-ab2cf21ff865ed3a

℘(A)∖{∅}\Pow{A} \setminus \{\emptyset\}

Read as: the power set of capital A set minus the set containing the empty set

Means: the power set of capital A set minus the set containing the empty set

1 occurrence in this chapter

Equation form expr-ac1ec3e15c4f6596

A≼β\cardle{A}{\beta}

Read as: capital A has cardinality at most that of beta

Means: capital A has cardinality at most that of beta

1 occurrence in this chapter

Equation form expr-adc48229342fddc7

B=ran(f)R={⟨f(α),f(β)⟩∈A×A:α∈β}.B &= \ran{f}\\ R &= \Setabs{\tuple{f(\alpha), f(\beta)} \in A \times A}{\alpha \in \beta}.

Read as: Source-ordered display. capital B equals the range of f. Then, capital R equals the set of the ordered pair f of alpha, then f of beta belongs to capital A times capital A such that alpha belongs to beta. End display

Means: Source-ordered display. capital B equals the range of f. Then, capital R equals the set of the ordered pair f of alpha, then f of beta belongs to capital A times capital A such that alpha belongs to beta. End display

1 occurrence in this chapter

Equation form expr-ae3660560a54702c

⇒\Rightarrow

Read as: implies

Means: implies

2 occurrences in this chapter

Equation form expr-afd16dfb3f8c66c7

Bn=An∖⋃i<nAi.B_n = A_n \setminus \bigcup_{i < n} A_i.

Read as: capital B subscript n equals capital A subscript n set minus the union over i is less than n of capital A subscript i

Means: capital B subscript n equals capital A subscript n set minus the union over i is less than n of capital A subscript i

1 occurrence in this chapter

Equation form expr-b041324b1af3b519

r∼sr \sim s

Read as: r is equivalent to s

Means: r is equivalent to s

1 occurrence in this chapter

Equation form expr-b36e04a95c1e7ed1

f:α→⟨A,R⟩f \colon \alpha \to \tuple{A, R}

Read as: f colon alpha maps to the ordered pair capital A, then capital R

Means: f colon alpha maps to the ordered pair capital A, then capital R

1 occurrence in this chapter

Equation form expr-b3974d5a37cdc7ef

fn:ω→Anf_n \colon \omega \to A_n

Read as: f subscript n colon omega maps to capital A subscript n

Means: f subscript n colon omega maps to capital A subscript n

1 occurrence in this chapter

Equation form expr-b3ca23795ebd1e38

{∅}\{\emptyset\}

Read as: the set containing the empty set

Means: the set containing the empty set

1 occurrence in this chapter

Equation form expr-b4668eacf7fe6b79

A≈B\cardeq{A}{B}

Read as: capital A is equinumerous with capital B

Means: capital A is equinumerous with capital B

1 occurrence in this chapter

Equation form expr-b6c9f7eeb5b5c2bf

α∉β\alpha \notin \beta

Read as: alpha does not belong to beta

Means: alpha does not belong to beta

1 occurrence in this chapter

Equation form expr-b76412237f06fe45

r1∼r2r_{1} \sim r_{2}

Read as: r subscript one is equivalent to r subscript two

Means: r subscript one is equivalent to r subscript two

1 occurrence in this chapter

Equation form expr-b8139cacceea2c7b

E={[r]∼:r∈S},E = \Setabs{\equivrep{r}{\sim}}{r \in \onesphere},

Read as: capital E equals the set of the equivalence class of r modulo is equivalent to such that r belongs to the unit circle

Means: capital E equals the set of the equivalence class of r modulo is equivalent to such that r belongs to the unit circle

1 occurrence in this chapter

Equation form expr-b880bf8b5c28c347

ℝ2\Real^2

Read as: the real numbers superscript two

Means: the real numbers superscript two

1 occurrence in this chapter

Equation form expr-bb2b778a1fa39d34

γ=ord(Bb,Rb)\gamma = \ordtype{B_b, R_b}

Read as: gamma equals the order type of capital B subscript b, then capital R subscript b

Means: gamma equals the order type of capital B subscript b, then capital R subscript b

1 occurrence in this chapter

Equation form expr-bbb9f8aa8fec5523

12\nicefrac{1}{2}

Read as: the fraction one over two

Means: the fraction one over two

1 occurrence in this chapter

Equation form expr-be025f8b1d6a787b

B⊆RB \subseteq R

Read as: capital B is a subset of capital R

Means: capital B is a subset of capital R

1 occurrence in this chapter

Equation form expr-beb14de5f7e22139

AnA_n

Read as: capital A subscript n

Means: capital A subscript n

1 occurrence in this chapter

Equation form expr-c2be9d6383f714df

r∼siff(∃ρ∈R)ρ(r)=s.r \sim s \emph{ iff }(\exists \rho \in \rotationsgroup)\rho(r) = s.

Read as: r is equivalent to s applied to iff applied to there exists rho in the group of rotations of the unit circle, rho of r equals s

Means: r is equivalent to s applied to iff applied to there exists rho in the group of rotations of the unit circle, rho of r equals s

1 occurrence in this chapter

Equation form expr-c33ab542c60c1577

cn∈Ac_n \in A

Read as: c subscript n belongs to capital A

Means: c subscript n belongs to capital A

1 occurrence in this chapter

Equation form expr-c647837e26767b55

x∈dom(f)x \in \dom{f}

Read as: x belongs to the domain of f

Means: x belongs to the domain of f

1 occurrence in this chapter

Equation form expr-c761fdac01944508

s∈Ss \in \onesphere

Read as: s belongs to the unit circle

Means: s belongs to the unit circle

1 occurrence in this chapter

Equation form expr-c7cb12bc94ac6fce

β⋠A\cardnless{\beta}{A}

Read as: beta does not have smaller cardinality than capital A

Means: beta does not have smaller cardinality than capital A

1 occurrence in this chapter

Equation form expr-ca978112ca1bbdca

aa

Read as: a

Means: a

1 occurrence in this chapter

Equation form expr-cc53ddce897ae888

R1\rotationsgroup_{1}

Read as: the group of rotations of the unit circle subscript one

Means: the group of rotations of the unit circle subscript one

3 occurrences in this chapter

Equation form expr-ccc00843ec25cbbd

s=ρ1(r1)=ρ2(r1)s = \rho_1(r_1) = \rho_2(r_1)

Read as: s equals rho subscript one of r subscript one equals rho subscript two of r subscript one

Means: s equals rho subscript one of r subscript one equals rho subscript two of r subscript one

1 occurrence in this chapter

Equation form expr-cd0aa9856147b6c5

gg

Read as: g

Means: g

4 occurrences in this chapter

Equation form expr-cec49e36e9b61af5

1=μ(S)=∑ρ∈Rμ(ρ[C])=∑ρ∈Rr1 = \mu(\onesphere) = \sum_{\rho \in \rotationsgroup}\mu(\funimage{\rho}{C}) = \sum_{\rho \in \rotationsgroup}r

Read as: one equals mu of the unit circle equals the sum over rho belongs to the group of rotations of the unit circle of mu of the image of capital C under rho equals the sum over rho belongs to the group of rotations of the unit circle of r

Means: one equals mu of the unit circle equals the sum over rho belongs to the group of rotations of the unit circle of mu of the image of capital C under rho equals the sum over rho belongs to the group of rotations of the unit circle of r

1 occurrence in this chapter

Equation form expr-cefb20269b7a575e

A≉n\cardneq{A}{n}

Read as: capital A is not equinumerous with n

Means: capital A is not equinumerous with n

1 occurrence in this chapter

Equation form expr-cf7abb44235c7889

α≺℘(A)∖{∅}\cardless{\alpha}{\Pow{A} \setminus \{\emptyset\}}

Read as: alpha has smaller cardinality than the power set of capital A set minus the set containing the empty set

Means: alpha has smaller cardinality than the power set of capital A set minus the set containing the empty set

1 occurrence in this chapter

Equation form expr-d0f3742f10192466

A=g[α]A = \funimage{g}{\alpha}

Read as: capital A equals the image of alpha under g

Means: capital A equals the image of alpha under g

1 occurrence in this chapter

Equation form expr-d195ec6120311945

α=ord(B,R)\alpha = \ordtype{B, R}

Read as: alpha equals the order type of capital B, then capital R

Means: alpha equals the order type of capital B, then capital R

1 occurrence in this chapter

Equation form expr-d6ed8162dea38ef2

BnB_n

Read as: capital B subscript n

Means: capital B subscript n

3 occurrences in this chapter

Equation form expr-d74b4b2a4d4699c4

S=⋃ρ∈Rρ[C]=⋃ρ∈R1(ρ∘ρ)[C]\onesphere = \bigcup_{\rho \in \rotationsgroup}\funimage{\rho}{C} = \bigcup_{\rho \in \rotationsgroup_1}\funimage{(\comp{\rho}{\rho})}{C}

Read as: the unit circle equals the union over rho belongs to the group of rotations of the unit circle of the image of capital C under rho equals the union over rho belongs to the group of rotations of the unit circle subscript one of the image of capital C under open scope, rho composed with rho, close scope

Means: the unit circle equals the union over rho belongs to the group of rotations of the unit circle of the image of capital C under rho equals the union over rho belongs to the group of rotations of the unit circle subscript one of the image of capital C under open scope, rho composed with rho, close scope

1 occurrence in this chapter

Equation form expr-d9de7658dee09cb8

ρ∈R1\rho \in R_1

Read as: rho belongs to capital R subscript one

Means: rho belongs to capital R subscript one

1 occurrence in this chapter

Equation form expr-dabd3aff769f07eb

<<

Read as: is less than

Means: is less than

1 occurrence in this chapter

Equation form expr-dabec2cdc19c3167

ρ\rho

Read as: rho

Means: rho

1 occurrence in this chapter

Equation form expr-db267d26ea5710d1

β=ord(B,R)\beta = \ordtype{B, R}

Read as: beta equals the order type of capital B, then capital R

Means: beta equals the order type of capital B, then capital R

1 occurrence in this chapter

Equation form expr-dbbf6d482d9c3a56

tso(A,<)=[⟨X,R⟩:⟨A,<⟩≅⟨X,R⟩]\text{tso}(A, <) = [\tuple{X, R} : \ordeq{\tuple{A, <}}{\tuple{X, R}}]

Read as: the T S order type of capital A under the less than relation equals the Tarski Scott set of ordered pairs capital X and capital R of least possible rank such that the ordered pair capital A and the less than relation is order isomorphic to the ordered pair capital X and capital R

Means: the T S order type of capital A under the less than relation equals the Tarski Scott set of ordered pairs capital X and capital R of least possible rank such that the ordered pair capital A and the less than relation is order isomorphic to the ordered pair capital X and capital R

1 occurrence in this chapter

Equation form expr-dc11a64df075c117

s=ρ1(r1)=ρ2(r2)s = \rho_{1}(r_{1}) = \rho_{2}(r_{2})

Read as: s equals rho subscript one of r subscript one equals rho subscript two of r subscript two

Means: s equals rho subscript one of r subscript one equals rho subscript two of r subscript two

1 occurrence in this chapter

Equation form expr-de1d196dbd462bbd

An⊆AA_n \subseteq A

Read as: capital A subscript n is a subset of capital A

Means: capital A subscript n is a subset of capital A

1 occurrence in this chapter

Equation form expr-df7e70e5021544f4

BB

Read as: capital B

Means: capital B

5 occurrences in this chapter

Equation form expr-e237f4977b9c4b9c

μ(X)=μ(Y)\mu(X) = \mu(Y)

Read as: mu of capital X equals mu of capital Y

Means: mu of capital X equals mu of capital Y

1 occurrence in this chapter

Equation form expr-e255e9267d6b72fd

C∩xC \cap x

Read as: capital C intersected with x

Means: capital C intersected with x

1 occurrence in this chapter

Equation form expr-e2972a7b24ee649f

ρ1[C]∩ρ2[C]=∅\funimage{\rho_1}{C} \cap \funimage{\rho_2}{C} = \emptyset

Read as: the image of capital C under rho subscript one intersected with the image of capital C under rho subscript two equals the empty set

Means: the image of capital C under rho subscript one intersected with the image of capital C under rho subscript two equals the empty set

1 occurrence in this chapter

Equation form expr-e349851823caf578

ZFC\ZFC

Read as: set theory Z F C

Means: set theory Z F C

4 occurrences in this chapter

Equation form expr-e403ac3df26602be

σ∘ρ=ρ∘σ\comp{\rho}{\sigma} = \comp{\sigma}{\rho}

Read as: sigma composed with rho equals rho composed with sigma

Means: sigma composed with rho equals rho composed with sigma

1 occurrence in this chapter

Equation form expr-e4d9b1f742f58655

r1,r2∈Cr_{1}, r_{2} \in C

Read as: r subscript one, then r subscript two belongs to capital C

Means: r subscript one, then r subscript two belongs to capital C

1 occurrence in this chapter

Equation form expr-e5eb2ea28a5ab876

⟨B,R⟩∈C\tuple{B, R} \in C

Read as: the ordered pair capital B, then capital R belongs to capital C

Means: the ordered pair capital B, then capital R belongs to capital C

1 occurrence in this chapter

Equation form expr-e7517b779c961363

14\nicefrac{1}{4}

Read as: the fraction one over four

Means: the fraction one over four

1 occurrence in this chapter

Equation form expr-e7ec977fc2e8350e

{x:A≈x}.\Setabs{x}{\cardeq{A}{x}}.

Read as: the set of x such that capital A is equinumerous with x

Means: the set of x such that capital A is equinumerous with x

1 occurrence in this chapter

Equation form expr-ea98652f597e123f

g∘f−1:A→B\comp{f^{-1}}{g} \colon A \to B

Read as: g composed with f superscript minus one colon capital A maps to capital B

Means: g composed with f superscript minus one colon capital A maps to capital B

1 occurrence in this chapter

Equation form expr-ec1ab83a0eb4c40c

r∈ℝr \in \Real

Read as: r belongs to the real numbers

Means: r belongs to the real numbers

1 occurrence in this chapter

Equation form expr-ec754c238c4f2a2b

⟨B,S⟩\tuple{B, S}

Read as: the ordered pair capital B, then capital S

Means: the ordered pair capital B, then capital S

1 occurrence in this chapter

Equation form expr-ee3b29601124f44d

[x:ϕ(x)][x : \phi(x)]

Read as: the Tarski Scott set of all x of least possible rank such that phi holds of x, or the empty set if there are none

Means: the Tarski Scott set of all x of least possible rank such that phi holds of x, or the empty set if there are none

1 occurrence in this chapter

Equation form expr-ee49d13dcfce4888

ρ1=ρ2\rho_1 = \rho_2

Read as: rho subscript one equals rho subscript two

Means: rho subscript one equals rho subscript two

1 occurrence in this chapter

Equation form expr-eec28c25460d2952

ω≼A\cardle{\omega}{A}

Read as: omega has cardinality at most that of capital A

Means: omega has cardinality at most that of capital A

2 occurrences in this chapter

Equation form expr-eec3ac3caa8bd397

(∀β∈α)(∀x⊆Vβ)¬ϕ(x)(\forall \beta \in \alpha)(\forall x \subseteq V_\beta)\lnot \phi(x)

Read as: for every beta in alpha, for every x is a subset of capital V subscript beta, not phi of x

Means: for every beta in alpha, for every x is a subset of capital V subscript beta, not phi of x

1 occurrence in this chapter

Equation form expr-f11aed8b3c261db5

B≼A\cardle{B}{A}

Read as: capital B has cardinality at most that of capital A

Means: capital B has cardinality at most that of capital A

2 occurrences in this chapter

Equation form expr-f3f3804480e8551a

β\beta

Read as: beta

Means: beta

1 occurrence in this chapter

Equation form expr-f51bc5408481a8a0

r∈Cr \in C

Read as: r belongs to capital C

Means: r belongs to capital C

1 occurrence in this chapter

Equation form expr-f65e5a0cb4d19a26

μ(σ[C])=r\mu(\funimage{\sigma}{C}) = r

Read as: mu of the image of capital C under sigma equals r

Means: mu of the image of capital C under sigma equals r

1 occurrence in this chapter

Equation form expr-f67ab10ad4e4c531

FF

Read as: capital F

Means: capital F

3 occurrences in this chapter

Equation form expr-f6ddcd23c958c7e1

#xFx\# x Fx

Read as: the number determined by x applied to Fx

Means: the number determined by x applied to Fx

1 occurrence in this chapter

Equation form expr-f75dd322f11ad636

A2A_2

Read as: capital A subscript two

Means: capital A subscript two

1 occurrence in this chapter

Equation form expr-f847d7618ed63aee

D2D_{2}

Read as: capital D subscript two

Means: capital D subscript two

2 occurrences in this chapter

Equation form expr-fa43237e30275465

ρ(r)=s\rho(r) = s

Read as: rho of r equals s

Means: rho of r equals s

1 occurrence in this chapter

Equation form expr-fcfbe991e8a25e0f

XnX_n

Read as: capital X subscript n

Means: capital X subscript n

1 occurrence in this chapter

Equation form expr-ff4be5f76635a0aa

ℶ1\beth_1

Read as: beth subscript one

Means: beth subscript one

1 occurrence in this chapter

Definition: Tarski--Scott

This source definition contains, in source order: phi of x; then the Tarski Scott set of all x of least possible rank such that phi holds of x, or the empty set if there are none; then x; then phi of x; then the empty set; then phi. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition two in this chapter

This source definition contains, in source order: capital A; then the T S cardinality of capital A equals the Tarski Scott set of all x of least possible rank that are equinumerous with capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma: in set theory Z F

This source lemma contains, in source order: set theory Z F; then capital A; then alpha; then alpha does not have smaller cardinality than capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math one in this chapter

This source display math contains, in source order: Source-ordered display. capital B equals the range of f. Then, capital R equals the set of the ordered pair f of alpha, then f of beta belongs to capital A times capital A such that alpha belongs to beta. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: in set theory Z F

This source theorem contains, in source order: set theory Z F; then capital A has cardinality at most that of capital B; then capital B has cardinality at most that of capital A; then capital A; then capital B. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition three in this chapter

This source definition contains, in source order: f; then f of x belongs to x; then x belongs to the domain of f; then f; then capital A; then f; then the domain of f equals capital A set minus the set containing the empty set. The complete surrounding source prose remains in the continuous listener stream.

Source

Axiom: Choice

This source axiom contains no delimiter-counted formula. Its complete source wording remains in the continuous listener stream.

Source

Theorem: in set theory Z F

This source theorem contains, in source order: set theory Z F. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math two in this chapter

This source display math contains, in source order: Source-ordered display. g of zero equals f of capital A. Then, g of alpha. End display is defined by cases. case one, stop!; if, capital A equals the image of alpha under g. case two, f of capital A set minus the image of alpha under g; otherwise. End cases. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma: in set theory Z minus

This source lemma contains, in source order: set theory Z minus. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition four in this chapter

This source definition contains no delimiter-counted formula. Its complete source wording remains in the continuous listener stream.

Source

Example one in this chapter

This source example contains, in source order: capital A; then omega has cardinality at most that of capital A; then capital A is equinumerous with n; then n belongs to omega; then set theory Z F C; then capital A; then the cardinality of capital A; then set theory Z minus plus Countable Choice; then capital A; then omega has cardinality at most that of capital A; then capital A is equinumerous with n; then n belongs to omega; then capital A is not equinumerous with n; then n belongs to omega; then n is less than omega; then capital A subscript n is a subset of capital A; then cardinal exponentiation of two to the power n; then capital A subscript zero, then capital A subscript one, then capital A subscript two, and so on; then n; then capital B subscript n equals capital A subscript n set minus the union over i is less than n of capital A subscript i; then Source-ordered display. the cardinality of the union over i is less than n of capital A subscript n is less than or equal to the cardinality of capital A subscript zero plus the cardinality of capital A subscript one. Then, equals one plus two plus , and so on plus two superscript n minus one. Then, equals two superscript n minus one. Then, is less than two superscript n equals the cardinality of capital A subscript n. End display; then capital B subscript n; then c subscript n; then capital B subscript n; then c subscript n equals c subscript m; then n equals m; then c subscript n belongs to capital A; then f of n equals c subscript n; then omega maps to capital A; then capital A subscript zero; then capital A subscript one; then capital A subscript two; then c subscript n; then capital B subscript n; then set theory Z F; then omega. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: in set theory Z minus plus Countable Choice

This source theorem contains, in source order: set theory Z minus plus Countable Choice; then capital A; then omega has cardinality at most that of capital A; then capital A is equinumerous with n; then n belongs to omega. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math three in this chapter

This source display math contains, in source order: Source-ordered display. the cardinality of the union over i is less than n of capital A subscript n is less than or equal to the cardinality of capital A subscript zero plus the cardinality of capital A subscript one. Then, equals one plus two plus , and so on plus two superscript n minus one. Then, equals two superscript n minus one. Then, is less than two superscript n equals the cardinality of capital A subscript n. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Example two in this chapter

This source example contains, in source order: set theory Z minus plus Countable Choice; then capital A subscript n; then n belongs to omega; then the union over n is less than omega of capital A subscript n; then capital A subscript n is not equal to the empty set; then n belongs to omega; then f subscript n colon omega maps to capital A subscript n; then f colon omega times omega maps to the union over n is less than omega of capital A subscript n; then f of m and n equals f subscript n of m; then omega times omega; then f; then f subscript zero; then f subscript one; then f subscript two; then beta belongs to the cardinal a; then f subscript beta; then set theory Z F; then beth subscript one; then aleph subscript zero; then aleph subscript zero. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: in set theory Z minus plus Countable Choice

This source theorem contains, in source order: set theory Z minus plus Countable Choice; then capital A subscript n; then n belongs to omega; then the union over n is less than omega of capital A subscript n. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: in set theory Z F

This source theorem contains, in source order: set theory Z F; then capital A; then capital C; then capital C intersected with x; then x belongs to capital A; then capital C; then capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Exercise one in this chapter

This source exercise contains no delimiter-counted formula. Its complete source wording remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Theorem: Banach--Tarski Paradox (in set theory Z F C)

This source theorem contains, in source order: set theory Z F C. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: Vitali's Paradox (in set theory Z F C)

This source theorem contains, in source order: set theory Z F C. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma three in this chapter

This source lemma contains, in source order: the group of rotations of the unit circle. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma four in this chapter

This source lemma contains, in source order: the group of rotations of the unit circle; then the group of rotations of the unit circle subscript one; then the group of rotations of the unit circle subscript two; then the group of rotations of the unit circle. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma five in this chapter

This source lemma contains, in source order: is equivalent to. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma six in this chapter

This source lemma contains, in source order: the unit circle equals the union over rho belongs to the group of rotations of the unit circle of the image of capital C under rho. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma seven in this chapter

This source lemma contains, in source order: rho subscript one is not equal to rho subscript two; then the image of capital C under rho subscript one intersected with the image of capital C under rho subscript two equals the empty set. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma eight in this chapter

This source lemma contains, in source order: the unit circle; then capital D subscript one; then capital D subscript two; then capital D subscript one; then the unit circle; then capital D subscript two. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math four in this chapter

This source display math contains, in source order: capital D subscript one equals the union over rho belongs to the group of rotations of the unit circle subscript one of the image of capital C under rho capital D subscript two equals the union over rho belongs to the group of rotations of the unit circle subscript two of the image of capital C under rho. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: Hausdorff's Paradox (in set theory Z F C)

This source theorem contains, in source order: set theory Z F C. The complete surrounding source prose remains in the continuous listener stream.

Source

Corollary: Vitali

This source corollary contains, in source order: mu; then mu of the unit circle equals one; then mu of capital X equals mu of capital Y; then capital X; then capital Y; then the image of capital C under rho; then rho belongs to the group of rotations of the unit circle. The complete surrounding source prose remains in the continuous listener stream.

Source

Cross-reference reference-001560

chapter “Cardinals”

Source occurrence

Cross-reference reference-001561

chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001562

definition one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001563

chapter “The Size of Sets”

Source occurrence

Cross-reference reference-001564

section “Appendix: Hume's Principle” in chapter “Cardinals”

Source occurrence

Cross-reference reference-001565

theorem three in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001566

definition one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001567

the lemma on Cardinals Behave Right

Source occurrence

Cross-reference reference-001568

the lemma on Size Powersettwo Exp

Source occurrence

Cross-reference reference-001569

Michael Potter (2004), chs. 9–12

Source occurrence

Cross-reference reference-001570

Friedrich Hartogs (1915)

Source occurrence

Cross-reference reference-001571

lemma four in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001572

theorem five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001573

corollary four in chapter “Ordinals”

Source occurrence

Cross-reference reference-001574

lemma three in chapter “Ordinals”

Source occurrence

Cross-reference reference-001575

item 1 of theorem “in set theory Z F” in chapter “Choice”

Source occurrence

Cross-reference reference-001576

item 2 of theorem “in set theory Z F” in chapter “Choice”

Source occurrence

Cross-reference reference-001577

item 1 of theorem “in set theory Z F” in chapter “Choice”

Source occurrence

Cross-reference reference-001578

theorem five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001579

proposition five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001580

item 2 of theorem “in set theory Z F” in chapter “Choice”

Source occurrence

Cross-reference reference-001581

item 1 of theorem “in set theory Z F” in chapter “Choice”

Source occurrence

Cross-reference reference-001582

lemma “in set theory Z F” in chapter “Choice”

Source occurrence

Cross-reference reference-001583

item 2 of theorem “in set theory Z F” in chapter “Choice”

Source occurrence

Cross-reference reference-001584

theorem two in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001585

1883

Source occurrence

Cross-reference reference-001586

Michael Potter

Source occurrence

Cross-reference reference-001587

2004, p. 243

Source occurrence

Cross-reference reference-001588

1904

Source occurrence

Cross-reference reference-001589

theorem “General Recursion” in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001590

lemma “in set theory Z F” in chapter “Choice”

Source occurrence

Cross-reference reference-001591

the proposition on pairsconsequences

Source occurrence

Cross-reference reference-001592

Michael Potter (2004), §9.4

Source occurrence

Cross-reference reference-001593

theorem one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001594

Richard Dedekind (1888)

Source occurrence

Cross-reference reference-001595

Paul J. Cohen (1966), p. 138

Source occurrence

Cross-reference reference-001596

1878

Source occurrence

Cross-reference reference-001597

proposition four in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001598

proposition “Enumerability of pairs of natural numbers” in chapter “The Size of Sets”

Source occurrence

Cross-reference reference-001599

proposition four in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001600

Solomon Feferman and Azriel Levy (1963)

Source occurrence

Cross-reference reference-001601

(Bertrand Russell, 1919, p. 126)

Source occurrence

Cross-reference reference-001602

section “The Story in More Detail” in chapter “Steps towards Z”

Source occurrence

Cross-reference reference-001603

theorem “in set theory Z F” in chapter “Choice”

Source occurrence

Cross-reference reference-001604

Michael Potter (2004), pp. 242–3

Source occurrence

Cross-reference reference-001605

section “The Story in More Detail” in chapter “Steps towards Z”

Source occurrence

Cross-reference reference-001606

2004, §14.8

Source occurrence

Cross-reference reference-001607

section “Extrinsic Considerations about Replacement” in chapter “Replacement”

Source occurrence

Cross-reference reference-001608

Stefan Banach and Alfred Tarski (1924)

Source occurrence

Cross-reference reference-001609

Grzegorz Tomkowicz and Stan Wagon (2016), Theorem 3.12

Source occurrence

Cross-reference reference-001610

Grzegorz Tomkowicz and Stan Wagon (2016), 303

Source occurrence

Cross-reference reference-001611

Michael Potter (2004), 276–7

Source occurrence

Cross-reference reference-001612

Tom Weston (2003), 16

Source occurrence

Cross-reference reference-001613

Grzegorz Tomkowicz and Stan Wagon (2016), 31, 308–9

Source occurrence

Cross-reference reference-001614

section “Pathologies” in chapter “History and Mythology of Set Theory”

Source occurrence

Cross-reference reference-001615

section “Cantor on the Line and the Plane” in chapter “History and Mythology of Set Theory”

Source occurrence

Cross-reference reference-001616

section “Pathologies” in chapter “History and Mythology of Set Theory”

Source occurrence

Cross-reference reference-001617

section “Appendix: Hilbert's Space-filling Curves” in chapter “History and Mythology of Set Theory”

Source occurrence

Cross-reference reference-001618

Raphael Robinson (1947)

Source occurrence

Cross-reference reference-001619

Grzegorz Tomkowicz and Stan Wagon (2016), pp. 66–7

Source occurrence

Cross-reference reference-001620

Tom Weston (2003)

Source occurrence

Cross-reference reference-001621

Grzegorz Tomkowicz and Stan Wagon (2016)

Source occurrence

Cross-reference reference-001622

1905

Source occurrence

Cross-reference reference-001623

Tom Weston (2003)

Source occurrence

Cross-reference reference-001624

theorem “Vitali's Paradox (in set theory Z F C)” in chapter “Choice”

Source occurrence

Cross-reference reference-001625

lemma three in chapter “Choice”

Source occurrence

Cross-reference reference-001626

lemma six in chapter “Choice”

Source occurrence

Cross-reference reference-001627

proposition four in chapter “Cardinal Arithmetic”

Source occurrence

Cross-reference reference-001628

lemma four in chapter “Choice”

Source occurrence

Cross-reference reference-001629

lemma six in chapter “Choice”

Source occurrence

Cross-reference reference-001630

lemma seven in chapter “Choice”

Source occurrence

Cross-reference reference-001631

lemma four in chapter “Choice”

Source occurrence

Cross-reference reference-001632

lemma six in chapter “Choice”

Source occurrence

Cross-reference reference-001633

lemma four in chapter “Choice”

Source occurrence

Cross-reference reference-001634

Raphael Robinson (1947)

Source occurrence

Cross-reference reference-001635

Grzegorz Tomkowicz and Stan Wagon (2016), Theorem 5.2

Source occurrence

Cross-reference reference-001636

Tom Weston (2003), p. 3

Source occurrence

Cross-reference reference-001637

Grzegorz Tomkowicz and Stan Wagon (2016), Theorem 2.1

Source occurrence

Cross-reference reference-001638

lemma eight in chapter “Choice”

Source occurrence

Cross-reference reference-001639

(Felix Hausdorff, 1914)

Source occurrence

Cross-reference reference-001640

Tom Weston (2003), p. 16

Source occurrence

Cross-reference reference-001641

section “The Banach--Tarski Paradox” in chapter “Choice”

Source occurrence

Cross-reference reference-001642

section “The Banach--Tarski Paradox” in chapter “Choice”

Source occurrence

Cross-reference reference-001643

lemma eight in chapter “Choice”

Source occurrence

Cross-reference reference-001644

lemma six in chapter “Choice”

Source occurrence

Cross-reference reference-001645

lemma seven in chapter “Choice”

Source occurrence

Source disclosures